← all cardsTopic 8 · Oscillations · Equations & the two systems
Speed at any displacement
● VERBATIM — examined word-for-wordThe timing-free workhorse — no t needed, only position. Sign gives direction.
v = ±ω√(x₀² − x²)What this actually means
Given a position, this hands you the speed with no timing information at all. It is energy conservation in disguise: kinetic energy is whatever total energy is left after potential.
The ± is direction: the particle passes each x twice per cycle, once moving each way.
Two-position problems: write the formula at both positions and divide; ω or x₀ cancels and the unknowns fall out without any clock.
Prove it — watch it be true
- Pause the motion and read x and v off the aligned graphs
- Compute ω√(x₀² − x²) by hand and compare with the v reading
- Check the endpoints: v = 0 at x = ±x₀, v = ωx₀ at x = 0